Ungrouping objects connected via a vertex
Your group is a single object. You can select the object, go into edit mode, then separate parts of the mesh into separate objects.
From Separating Meshes - Blender Documentation

In edit mode press the hotkey 'P'. Then select one of the option:
Selected

This option separates the selection to a new object.

All Loose Parts

Separates the mesh in its unconnected parts.

By Material

Creates separate mesh objects for each material.


If the parts are unconnected in edit mode (as you stated), use Separate &gt; All Loose Parts.

They seem to be connected to random vertices at the grid origin.

If the parts are connected, You can try to can try to delete the vertices/edges before separating the mesh by loose parts. Otherwise you will have to select the part of the mesh, which you want to separate and separate by selected. (Useful:  'L' [Select Linked])